Thin elastic beams: The variational approach to St. Venant’s problem

Abstract

The behaviour of a linearly elastic homogeneous and isotropic beam bounded by a cylindrical surface (the lateral surface of the beam), by a pair of planes normal to the lateral surface (the bases of the cylinder) and subjected both to body forces and to surface tractions is well known in the literature as the St. Venant’s problem or some of its generalizations. From a mathematical point of view we are concerned with a problem of calculus of variations, that is, we have to minimize, in a class of suitable competing functions, a functional of the type